Nuprl Lemma : univ-trans_wf
∀[G:j⊢]. ∀[T:{G.𝕀 ⊢ _:c𝕌}]. (univ-trans(G;T) ∈ {G ⊢ _:((decode(T))[0(𝕀)] ⟶ (decode(T))[1(𝕀)])})
Proof
Definitions occuring in Statement :
univ-trans: univ-trans(G;T),
universe-decode: decode(t),
cubical-universe: c𝕌,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
univ-trans: univ-trans(G;T),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x]
Lemmas referenced :
transprt-fun_wf,
universe-decode_wf,
cube-context-adjoin_wf,
interval-type_wf,
comp-op-to-comp-fun_wf2,
cubical_set_cumulativity-i-j,
universe-comp-op_wf,
istype-cubical-universe-term,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
universeIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[T:\{G.\mBbbI{} \mvdash{} \_:c\mBbbU{}\}]. (univ-trans(G;T) \mmember{} \{G \mvdash{} \_:((decode(T))[0(\mBbbI{})] {}\mrightarrow{} (decode(T))[1(\mBbbI{})])\})
Date html generated:
2020_05_20-PM-07_31_59
Last ObjectModification:
2020_04_29-PM-11_11_05
Theory : cubical!type!theory
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